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Advanced Factorization

Advanced factorization is a gateway to algebraic number theory, which mathematicians study in order to solve famous conjectures like Fermat's Last Theorem.

Algebraic Identities

         

The polynomial \[9-4x^2-20xy-25y^2\] can be factorized as \((3+ax+by)(3+cx+dy),\) where \(a,\) \(b,\) \(c\) and \(d\) are real numbers. What is the value of \(abcd\)?

What is the value of \(a+b+c\) if \[\begin{align} & x^2+8y^2-6xy-2yz+zx+4x-16y+4z \\ &= (x-ay+b)(x-cy+z)? \end{align}\]

If \(a\), \(b\) and \(c\) are real numbers and \[(x^2+2x)^2-79(x^2+2x)-80=(x+a)^2(x+b)(x+c)\] is an identity in \(x\), what is the value of \(a+b+c\)?

What is the value of \(a+b+c\) if \[\begin{align} & (x^2+3x)^2+10x^2+30x-56 \\ &= (x-1)(x+a)(x^2+bx+c)? \end{align} \]

What is the value of \(a+b+c\) if \[\begin{align} & x^2+5xy+4y^2+5x+23y-6 \\ &= (x+ay-b)(x+y+c)? \end{align} \]

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